document.write( "Question 114528: Given y=3x^2+6x \r
\n" ); document.write( "\n" ); document.write( "Please explain the process in steps. Thanks!\r
\n" ); document.write( "\n" ); document.write( "A.) find the y-intercept
\n" ); document.write( "B.) find the x-intercepts (if any)
\n" ); document.write( "c.) find the vertex
\n" ); document.write( "D.) Sketch the parabola. \n" ); document.write( "
Algebra.Com's Answer #83323 by MathLover1(20850)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "Given:
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\n" ); document.write( "\n" ); document.write( "A.) find the \"y-intercept\"…………\r
\n" ); document.write( "\n" ); document.write( "find the \"y-intercept\" we need to set \"x\" equal to \"0\"\r
\n" ); document.write( "\n" ); document.write( "\"y+=3x%5E2%2B6x+\"\r
\n" ); document.write( "\n" ); document.write( "\"y+=3%2A0%5E2%2B6%2A0+\"\r
\n" ); document.write( "\n" ); document.write( "\"y+=+0+\"\r
\n" ); document.write( "\n" ); document.write( "B.) find the \"x-intercepts\" (if any)\r
\n" ); document.write( "\n" ); document.write( "To find the \"x-intercept\", set \"y\" equal \"0\" and solve:\r
\n" ); document.write( "\n" ); document.write( "\"0+=3x%5E2%2B6x+\"\r
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\n" ); document.write( "\"3x%5E2%2B6x+=+0+\"……………. In this quadratic equation factor \"x\" out\r
\n" ); document.write( "\n" ); document.write( "\"x%283x%2B6%29+=+0+\"…………….find solutions\r
\n" ); document.write( "\n" ); document.write( "Above product will be equal to \"0\" if either one or both factors are equal to \"0\"\r
\n" ); document.write( "\n" ); document.write( "So \"first\"\"+solution\" is \"x=0\"\r
\n" ); document.write( "\n" ); document.write( "\"second\"\"+solution\" is \"3x%2B6=0\", and this will be true if \"3x=-6\"} => \"x=-6%2F3\"\r
\n" ); document.write( "\n" ); document.write( "Or \"x+=+-2\"\r
\n" ); document.write( "\n" ); document.write( "Therefore, \"x_intercepts\" are at \"x=0\" and \"x+=+-2\"\r
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\n" ); document.write( "\n" ); document.write( "c.) find the \"vertex\"\r
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Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

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\n" ); document.write( " \"y=3+x%5E2%2B6+x%2B0\" Start with the given equation
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\n" ); document.write( " \"y-0=3+x%5E2%2B6+x\" Subtract \"0\" from both sides
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\n" ); document.write( " \"y-0=3%28x%5E2%2B2x%29\" Factor out the leading coefficient \"3\"
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\n" ); document.write( " Take half of the x coefficient \"2\" to get \"1\" (ie \"%281%2F2%29%282%29=1\").
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\n" ); document.write( " Now square \"1\" to get \"1\" (ie \"%281%29%5E2=%281%29%281%29=1\")
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\n" ); document.write( " \"y-0=3%28x%5E2%2B2x%2B1-1%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"1\" does not change the equation
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\n" ); document.write( " \"y-0=3%28%28x%2B1%29%5E2-1%29\" Now factor \"x%5E2%2B2x%2B1\" to get \"%28x%2B1%29%5E2\"
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\n" ); document.write( " \"y-0=3%28x%2B1%29%5E2-3%281%29\" Distribute
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\n" ); document.write( " \"y-0=3%28x%2B1%29%5E2-3\" Multiply
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\n" ); document.write( " \"y=3%28x%2B1%29%5E2-3%2B0\" Now add \"0\" to both sides to isolate y
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\n" ); document.write( " \"y=3%28x%2B1%29%5E2-3\" Combine like terms
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\n" ); document.write( " Now the quadratic is in vertex form \"y=a%28x-h%29%5E2%2Bk\" where \"a=3\", \"h=-1\", and \"k=-3\". Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
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\n" ); document.write( " Check:
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\n" ); document.write( " Notice if we graph the original equation \"y=3x%5E2%2B6x%2B0\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3x%5E2%2B6x%2B0%29\" Graph of \"y=3x%5E2%2B6x%2B0\". Notice how the vertex is (\"-1\",\"-3\").
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\n" ); document.write( " Notice if we graph the final equation \"y=3%28x%2B1%29%5E2-3\" we get:
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\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3%28x%2B1%29%5E2-3%29\" Graph of \"y=3%28x%2B1%29%5E2-3\". Notice how the vertex is also (\"-1\",\"-3\").
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\n" ); document.write( " So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.
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\n" ); document.write( "\n" ); document.write( "D.) Sketch the parabola.\r
\n" ); document.write( "\n" ); document.write( "See \"c.%29\"
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